Integer Moments Research Articles

The Lognormal Distribution Is Characterized by Its Integer Moments

The lognormal moment sequence is considered. Using the fractional moments technique, it is first proved that the lognormal has the largest differential entropy among the infinite positively supported probability densities with the same lognormal-moments. Then, relying on previous theoretical results on entropy convergence obtained by the authors concerning the indeterminate Stieltjes moment problem, the lognormal distribution is accurately reconstructed by the maximum entropy technique using only its integer moment sequence, although it is not uniquely determined by moments.

On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein–Uhlenbeck processes

We establish sufficient conditions for the existence, and derive explicit formulas for the κ’th moments, κ≥1, of Markov modulated generalized Ornstein–Uhlenbeck processes as well as their stationary distributions. In particular, the running mean, the autocovariance function, and integer moments of the stationary distribution are derived in terms of the characteristics of the driving Markov additive process.Our derivations rely on new general results on moments of Markov additive processes and (multidimensional) integrals with respect to Markov additive processes.

On the gamma difference distribution

The gamma difference distribution is defined as the difference of two independent gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear differential equation specification of the probability density function is given. This is used to derive a Stein-type differential identity relating to the expectation with respect to the gamma difference distribution of a general twice differentiable function g(x). Choosing g(x)=xk gives a second order recurrence for the positive integer moments, which are also shown to permit evaluations in terms of 2F1 hypergeometric polynomials. A hypergeometric function evaluation is given for the absolute continuous moments. Specialising the gamma difference distribution gives the variance gamma distribution. Results of the type obtained herein have previously been obtained for this distribution, allowing for comparisons to be made.

Indeterminate Stieltjes Moment Problem: Entropy Convergence

The aim of this paper is to consider the indeterminate Stieltjes moment problem together with all its probability density functions that have the positive real or the entire real axis as support. As a consequence of the concavity of the entropy function in both cases, there is one such density that has the largest entropy: we call it fhmax, the largest entropy density. We will prove that the Jaynes maximum entropy density (MaxEnt), constrained by an increasing number of integer moments, converges in entropy to the largest entropy density fhmax. Note that this kind of convergence implies convergence almost everywhere, with remarkable consequences in real applications in terms of the reliability of the results obtained by the MaxEnt approximation of the underlying unknown distribution, both for the determinate and the indeterminate case.

Probability Distributions Approximation via Fractional Moments and Maximum Entropy: Theoretical and Computational Aspects

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance of the analogous procedure based on integer moments. No attention has been paid to two formal aspects concerning fractional moments, such as conditions for the existence of the maximum entropy approximation based on them or convergence in entropy of this approximation to F. This paper aims to fill this gap by providing proofs of these two fundamental results. In fact, convergence in entropy can be involved in the optimal selection of the order of fractional moments for accelerating the convergence of the MaxEnt approximation to F, to clarify the entailment relationships of this type of convergence with other types of convergence useful in statistical applications, and to preserve some important prior features of the underlying F distribution.

Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force

The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force is characterized by explicit dynamics of their integer moments and by explicit relaxation of spectral properties towards their steady state. Besides the Ornstein–Uhlenbeck process with a Gaussian steady state, other representative examples of the Pearson family are the square root or the Cox–Ingersoll–Ross process converging towards the gamma distribution, the Jacobi process converging towards the beta distribution, the reciprocal gamma process (corresponding to an exponential functional of the Brownian motion) that converges towards the inverse-gamma distribution, the Fisher–Snedecor process and the Student process. The last three steady states display heavy tails. The goal of the present paper is to analyze the large deviation properties of these various diffusion processes in a unified framework. We first consider level 1 concerning time-averaged observables over a large time window T. We write the first rescaled cumulants for generic observables and identify specific observables whose large deviations can be explicitly computed from the dominant eigenvalue of the appropriate deformed generator. The explicit large deviations at level 2 concerning the time-averaged density are then used to analyze the statistical inference of model parameters from data on a very long stochastic trajectory in order to obtain the explicit rate function for the two inferred parameters of the Pearson linear force.

Branching Random Walks with One Particle Generation Center and Possible Absorption at Every Point

We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only the absorption of particles can occur. The asymptotic behavior of the integer moments of both the total number of particles and the number of particles at a lattice point is studied depending on the relationship between the model parameters. In the case of the existence of an isolated positive eigenvalue of the evolution operator of the average number of particles, a limit theorem is obtained on the exponential growth of both the total number of particles and the number of particles at a lattice point.

Derivation of Bessel function closed-form solutions in zero dimensional φ4-field theory

The integral ∫−∞∞e−x2−gx4dx is used as an introductory learning tool in the study of Quantum Field Theory and path integrals. Typically, it is analyzed via perturbation theory. Closed-form solutions have been quoted for which I could not find any derivation. Using a simple and elegant transformation, the close form solutions for the integral and its even positive integer moments can be obtained in terms of Bessel functions.

On moments of downward passage times for spectrally negative Lévy processes

AbstractThe existence of moments of first downward passage times of a spectrally negative Lévy process is governed by the general dynamics of the Lévy process, i.e. whether it is drifting to $+\infty$ , $-\infty$ , or oscillating. Whenever the Lévy process drifts to $+\infty$ , we prove that the $\kappa$ th moment of the first passage time (conditioned to be finite) exists if and only if the $(\kappa+1)$ th moment of the Lévy jump measure exists. This generalizes a result shown earlier by Delbaen for Cramér–Lundberg risk processes. Whenever the Lévy process drifts to $-\infty$ , we prove that all moments of the first passage time exist, while for an oscillating Lévy process we derive conditions for non-existence of the moments, and in particular we show that no integer moments exist.

Fractional characteristic functions, and a fractional calculus approach for moments of random variables

In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann–Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented.

On the Moments of the Partition Function of the Cbeta E Field

We obtain a combinatorial formula for the positive integer moments of the partition function of the Cbeta E_{N} field, or equivalently the moments of the moments of the characteristic polynomial of the Cbeta E_{N} ensemble. We then use this formula to establish the large N asymptotics of these moments in the “moment-supercritical” regime. A key role is played by Jack polynomials.

Moment tests of independent components

We propose simple specification tests for independent component analysis and structural vector autoregressions with non-Gaussian shocks that check the normality of a single shock and the potential cross-sectional dependence among several of them. Our tests compare the integer (product) moments of the shocks in the sample with their population counterparts. Importantly, we explicitly consider the sampling variability resulting from using shocks computed with consistent parameter estimators. We study the finite sample size of our tests in several simulation exercises and discuss some bootstrap procedures. We also show that our tests have non-negligible power against a variety of empirically plausible alternatives.

Moments of generalized Cauchy random matrices and continuous-Hahn polynomials

In this paper we prove that, after an appropriate rescaling, the sum of moments of an N × N Hermitian matrix H sampled according to the generalized Cauchy (also known as Hua–Pickrell) ensemble with parameter s > 0 is a continuous-Hahn polynomial in the variable k. This completes the picture of the investigation that began in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) where analogous results were obtained for the other three classical ensembles of random matrices, the Gaussian, the Laguerre and Jacobi. Our strategy of proof is somewhat different from the one in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) due to the fact that the generalized Cauchy is the only classical ensemble which has a finite number of integer moments. Our arguments also apply, with straightforward modifications, to the other three cases studied in (Cunden et al 2019 Commun. Math. Phys. 369 1091–45) as well. We finally obtain a differential equation for the one-point density function of the eigenvalue distribution of this ensemble and establish the large N asymptotics of the moments.

Log-concavity and log-convexity of moments of averages of i.i.d. random variables

AbstractWe show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).

Recruitment prediction for multicenter clinical trials based on a hierarchical Poisson–gamma model: Asymptotic analysis and improved intervals

We analyze predictions of future recruitment to a multicenter clinical trial based on a maximum-likelihood fitting of a commonly used hierarchical Poisson-gamma model for recruitments at individual centers. We consider the asymptotic accuracy of quantile predictions in the limit as the number of recruitment centers grows large and find that, in an important sense, the accuracy of the quantiles does not improve as the number of centers increases. When predicting the number of further recruits in an additional time period, the accuracy degrades as the ratio of the additional time to the census time increases, whereas when predicting the amount of additional time to recruit a further patients, the accuracy degrades as the ratio of to the number recruited up to the census period increases. Our analysis suggests an improved quantile predictor. Simulation studies verify that the predicted pattern holds for typical recruitment scenarios in clinical trials and verify the much improved coverage properties of prediction intervals obtained from our quantile predictor. In the process of extending the applicability of our methodology, we show that in terms of the accuracy of all integer moments it is always better to approximate the sum of independent gamma random variables by a single gamma random variable matched on the first two moments than by the moment-matched Gaussian available from the central limit theorem.

Uncertainty analysis of rock tunnel based on fractional moment and dimensional reduction method

In this study, fractional moment was adopted to estimate the uncertainty in a tunnel excavated in rock. The multiplicative dimensional reduction method (M-DRM) was adopted to approximate the relationship between random variables and response of the tunnel and to predict its behavior. Integer and non-integer (fractional) moments were estimated by Gaussian quadrature based on M-DRM. The maximum entropy principle was also used to estimate the probability distribution of the tunnel response using the optimization technology of Excel Solver based on fractional moment. The example of a circular tunnel in a rock mass subjected to hydrostatic stress is given. This demonstrates the uncertainty analysis procedure applied to the displacements and plastic zone induced by the excavation, for different rock-support pressures and far-field stresses. Statistical moment, standard deviation and probability distribution of the tunnel response all agree well with Monte Carlo simulation (MCS). M-DRM closely approximates the tunnel response, and improves the efficiency of the uncertainty analysis. The proposed method estimates the probability distribution based on frictional moment without any prior knowledge. The results show that the proposed uncertainty analysis method is robust, efficient and precise, and is suitable for a complex practical engineering system as encountered in rock engineering.

Moments of Moments and Branching Random Walks

We calculate, for a branching random walk X_n(l) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable frac{1}{2^{n}}sum _{l=1}^{2^n}e^{2beta X_n(l)}, for beta in {mathbb {R}}. We obtain explicit formulae for the first few moments for finite n. In the limit nrightarrow infty , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.

A maximum entropy model with fractional moments for probability density function estimation of wind pressures on low-rise building

Due to the fickle distribution characteristics, good approximation of the wind pressure probability density function for an entire roof is challenging, especially for tail region. Maximum entropy model (MEM) can theoretically generate the least biased distribution. However, classical MEM may lead to unreliable estimates of the corresponding integer moments. This study developed a new maximum entropy model with fractional moments. There are several important features: A new translation function is introduced, thus allowing negative data to be modeled. For a good bias-variance trade-off, the constraint number M was fixed as four. After estimating the initial value of the Lagrange multiplier λ by a computationally simple linear equation system, a simple search was carried out to find a better solution. The generalized pattern search was adopted for determining the global optimal solution for the fractional moment orders α. The performance was benchmarked through typical field measurement data of wind pressures on the roof of a low-rise building under typhoon conditions. Compared with common models, the proposed method had better performance and more stable results for the dataset examined for the whole roof. This method is also beneficial for peak value evaluation and the simulation of wind pressures.

Expected Logarithm and Negative Integer Moments of a Noncentral χ2-Distributed Random Variable.

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral -distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

Rethinking jets with energy correlators: Tracks, resummation, and analytic continuation

We introduce an infinite set of jet substructure observables, derived as projections of $N$-point energy correlators, that are both convenient for experimental studies and maintain remarkable analytic properties derived from their representations in terms of a finite number of light ray operators. We show that these observables can be computed using tracking or charge information with a simple reweighting by integer moments of non-perturbative track or fragmentation functions. Our results for the projected $N$-point correlators are analytic functions of $N$, allowing us to derive resummed results to next-to-leading logarithmic accuracy for all $N$. We analytically continue our results to non-integer values of $N$ and define a corresponding analytic continuation of the observable, which we term a $\nu$-correlator, that can be measured on jets of hadrons at the LHC. This enables observables that probe the leading twist collinear dynamics of jets to be placed into a single analytic family, which we hope will lead to new insights into jet substructure.